(76-4x)(10+2x)=1080 eching

$$(76-4x)(10+2x)=1080$$

x uchun yechish

$$x=4$$

$$x=10$$

Kvadrat tenglamalaridan foydalanish qoidalari

76-4x ga 10+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$760+112x-8x^{2}=1080$$

Ikkala tarafdan 1080 ni ayirish.

$$760+112x-8x^{2}-1080=0$$

-320 olish uchun 760 dan 1080 ni ayirish.

$$-320+112x-8x^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-8x^{2}+112x-320=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 112 ni b va -320 ni c bilan almashtiring.

$$x=\frac{-112±\sqrt{112^{2}-4\left(-8\right)\left(-320\right)}}{2\left(-8\right)}$$

112 kvadratini chiqarish.

$$x=\frac{-112±\sqrt{12544-4\left(-8\right)\left(-320\right)}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-112±\sqrt{12544+32\left(-320\right)}}{2\left(-8\right)}$$

32 ni -320 marotabaga ko'paytirish.

$$x=\frac{-112±\sqrt{12544-10240}}{2\left(-8\right)}$$

12544 ni -10240 ga qo'shish.

$$x=\frac{-112±\sqrt{2304}}{2\left(-8\right)}$$

2304 ning kvadrat ildizini chiqarish.

$$x=\frac{-112±48}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{-112±48}{-16}$$

x=\frac{-112±48}{-16} tenglamasini yeching, bunda ± musbat. -112 ni 48 ga qo'shish.

$$x=-\frac{64}{-16}$$

-64 ni -16 ga bo'lish.

$$x=4$$

x=\frac{-112±48}{-16} tenglamasini yeching, bunda ± manfiy. -112 dan 48 ni ayirish.

$$x=-\frac{160}{-16}$$

-160 ni -16 ga bo'lish.

$$x=10$$

Tenglama yechildi.

$$x=4 x=10$$

Kvadratni yakunlash bo'yicha qadamlar

76-4x ga 10+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$760+112x-8x^{2}=1080$$

Ikkala tarafdan 760 ni ayirish.

$$112x-8x^{2}=1080-760$$

320 olish uchun 1080 dan 760 ni ayirish.

$$112x-8x^{2}=320$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-8x^{2}+112x=320$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}+112x}{-8}=\frac{320}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{112}{-8}x=\frac{320}{-8}$$

112 ni -8 ga bo'lish.

$$x^{2}-14x=\frac{320}{-8}$$

320 ni -8 ga bo'lish.

$$x^{2}-14x=-40$$

-14 ni bo‘lish, x shartining koeffitsienti, 2 ga -7 olish uchun. Keyin, -7 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-14x+\left(-7\right)^{2}=-40+\left(-7\right)^{2}$$

-7 kvadratini chiqarish.

$$x^{2}-14x+49=-40+49$$

-40 ni 49 ga qo'shish.

$$x^{2}-14x+49=9$$

x^{2}-14x+49 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-7\right)^{2}=9$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{9}$$

Qisqartirish.

$$x-7=3 x-7=-3$$

7 ni tenglamaning ikkala tarafiga qo'shish.

$$x=10 x=4$$